How do I make a round hay roof?
Image as reference, proportional editing/sculpting, and tip thickness
As you have a reference image, I think the best way to start the modeling is blocking out of the shape right onto the reference image to ensure the similarity of the proportions.
Keep the topology tidy in this phase as long as possible. In my test file I started by extruding and scaling a circle on the Z-axis while keeping an image-type empty behind. Don't worry about the a-symmetry of the reference object, just focus on the overall proportions.
For the tip I went for a grid fill, followed by some smooth vertex operator iterations.

The shape has now some kind of radial symmetry that we would like to avoid. I took advantage of the proportional editing tool to grab the vertices around and various sculpt tools to smooth the shape (or to bump out some parts).
 
Add to the roof a hair particle system, set the right amount of particles, comb the hairs toward the bottom (in particle edit mode) and use weight paint to adjust the density and length.

Once satisfied with the distribution, start tweaking the parameters of the individual hairs. I went for a long stick type by setting the Tip section equal to the Root section. You can find more info about how to control the shape of the strand in the Blender Manual page:


Finish by setting the Children particles type to one of the proposed modes (in this case I went for Interpolated) to increase the amount of particles.
Play with other parameters till you reach the right look for your job. Here's a possible result:

Note: this is just a presentation of the workflow and the tools I would use for the job. Each one probably requires some time and further research in order to be mastered.